• Korean Journal of Mathematics
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• v.17 no.1
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• pp.1-13
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• 2009

For an isometric immersion $f:M{\rightarrow}{\bar{M}}$ of Finsler manifolds M into $\bar{M}$, we compare the intrinsic Chern connection on M and the induced connection on M: We find the conditions for them to coincide and generalize the equations of Gauss, Ricci and Codazzi to Finsler submanifolds. In case the ambient space is a locally Minkowskian Finsler manifold, we simplify the above equations.

• Communications of the Korean Mathematical Society
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• v.17 no.1
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• pp.57-69
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• 2002

We study a unifying method to obtain nondegenerate isometric immersions of manifolds of constant sectional curvatures c with flat normal bundles into the space forms of curvature c for c= -1,0,1.

• Kyungpook Mathematical Journal
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• v.25 no.1
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• pp.77-82
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• 1985

• Communications of the Korean Mathematical Society
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• v.21 no.4
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• pp.729-737
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• 2006

We study Ribaucour transformations on nondegenerate local isometric immersions of Riemannian space forms into Lorentzian space forms with flat normal bundles. They can be explained by dressing actions on the solution space of Lorentzian Grassmannian systems.

• Kyungpook Mathematical Journal
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• v.25 no.2
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• pp.155-160
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• 1985

• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.237-249
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• 2002

We study nondegenerate immersions of Riemannian manifolds of constant sectional curvatures into Lorentzian manifolds of the same constant sectional curvatures with flat normal bundles. We also give a method to produce such immersions using the so-called Grassmannian system. .

• Journal of the Korean Mathematical Society
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• v.33 no.1
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• pp.169-181
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• 1996

Let $x : M^n \longrightarrow E^m$ be an isometric immersion of an n-dimensional connected Riemannian manifold into the m-dimensional Euclidean space. Then the metric tensor on $M^n$ is naturally induced from that of $E^m$.

• Bulletin of the Korean Mathematical Society
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• v.47 no.4
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• pp.767-775
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• 2010

Let ${\varphi}\;:\;(M^n,\;F)\;{\rightarrow}\;(\overline{M}^{n+p},\;\overline{F})$ be an isometric immersion from a Finsler manifold to a Finsler manifold. In this paper, we shall obtain the Gauss and Codazzi equations with respect to the Chern connection on submanifolds M, by which we prove that if M is a weakly totally geodesic submanifold of $\overline{M}$, then flag curvature of M equals flag curvature of $\overline{M}$.

• Bulletin of the Korean Mathematical Society
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• v.51 no.3
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• pp.823-829
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• 2014

Ruled submanifolds in Euclidean space satisfying some algebraic equations concerning the Laplace operator related to the isometric immersion and Gauss map are studied. Cylinders over a finite type curve or generalized helicoids are characterized with such algebraic equations.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1651-1670
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• 2016

In this paper, we investigate exponentially biharmonic maps u : (M, g) ${\rightarrow}$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that if $\int_{M}e^{\frac{p{\mid}r(u){\mid}^2}{2}{\mid}{\tau}(u){\mid}^pdv_g$ < ${\infty}$ ($p{\geq}2$), $\int_{M}{\mid}{\tau}(u){\mid}^2dv_g$ < ${\infty}$ and $\int_{M}{\mid}d(u){\mid}^2dv_g$ < ${\infty}$, then u is harmonic. When u is an isometric immersion, we get that if $\int_{M}e^{\frac{pm^2{\mid}H{\mid}^2}{2}}{\mid}H{\mid}^qdv_g$ < ${\infty}$ for 2 ${\leq}$ p < ${\infty}$ and 0 < q ${\leq}$ p < ${\infty}$, then u is minimal. We also obtain that any weakly convex exponentially biharmonic hypersurface in space form N(c) with $c{\leq}0$ is minimal. These results give affirmative partial answer to conjecture 3 (generalized Chen's conjecture for exponentially biharmonic submanifolds).